Linearly constrained quadratic program












0












$begingroup$


I have the following quadratic program



$$min_x x^TAx qquad text{s.t} quad Ax in [a,b]^m$$



where matrix $A$ is positive semidefinite, and is similar both the objective function and in the constraint. I would like to solve this problem for large-scale input matrix $A$ and am looking for a swift solution.



Is there any specific recommendation from different optimization methods such as projection gradient, proximal gradient, trust region active set, so forth? Any comparison between these methods (and others) are appreciated.






optimization convex-optimization numerical-optimization quadratic-programming







share|cite|improve this question











$endgroup$












  • $begingroup$
    Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:04






  • 1




    $begingroup$
    How large is $A$? Is it sparse or does it have any other structure?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:16










  • $begingroup$
    That is right, Ax is vector! I edited the question.
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 9:59










  • $begingroup$
    The only thing we know is that A is positive semi definite! Nothing more is available
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:00






  • 1




    $begingroup$
    @LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
    $endgroup$
    – littleO
    Dec 16 '18 at 20:56


















0












$begingroup$


I have the following quadratic program



$$min_x x^TAx qquad text{s.t} quad Ax in [a,b]^m$$



where matrix $A$ is positive semidefinite, and is similar both the objective function and in the constraint. I would like to solve this problem for large-scale input matrix $A$ and am looking for a swift solution.



Is there any specific recommendation from different optimization methods such as projection gradient, proximal gradient, trust region active set, so forth? Any comparison between these methods (and others) are appreciated.






optimization convex-optimization numerical-optimization quadratic-programming







share|cite|improve this question











$endgroup$












  • $begingroup$
    Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:04






  • 1




    $begingroup$
    How large is $A$? Is it sparse or does it have any other structure?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:16










  • $begingroup$
    That is right, Ax is vector! I edited the question.
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 9:59










  • $begingroup$
    The only thing we know is that A is positive semi definite! Nothing more is available
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:00






  • 1




    $begingroup$
    @LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
    $endgroup$
    – littleO
    Dec 16 '18 at 20:56
















0












0








0


2



$begingroup$


I have the following quadratic program



$$min_x x^TAx qquad text{s.t} quad Ax in [a,b]^m$$



where matrix $A$ is positive semidefinite, and is similar both the objective function and in the constraint. I would like to solve this problem for large-scale input matrix $A$ and am looking for a swift solution.



Is there any specific recommendation from different optimization methods such as projection gradient, proximal gradient, trust region active set, so forth? Any comparison between these methods (and others) are appreciated.






optimization convex-optimization numerical-optimization quadratic-programming







share|cite|improve this question











$endgroup$




I have the following quadratic program



$$min_x x^TAx qquad text{s.t} quad Ax in [a,b]^m$$



where matrix $A$ is positive semidefinite, and is similar both the objective function and in the constraint. I would like to solve this problem for large-scale input matrix $A$ and am looking for a swift solution.



Is there any specific recommendation from different optimization methods such as projection gradient, proximal gradient, trust region active set, so forth? Any comparison between these methods (and others) are appreciated.






optimization convex-optimization numerical-optimization quadratic-programming




optimization convex-optimization numerical-optimization quadratic-programming






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 20 at 16:04









Rodrigo de Azevedo

13k41958




13k41958










asked Dec 16 '18 at 8:05









Majid MohammadiMajid Mohammadi

275




275












  • $begingroup$
    Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:04






  • 1




    $begingroup$
    How large is $A$? Is it sparse or does it have any other structure?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:16










  • $begingroup$
    That is right, Ax is vector! I edited the question.
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 9:59










  • $begingroup$
    The only thing we know is that A is positive semi definite! Nothing more is available
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:00






  • 1




    $begingroup$
    @LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
    $endgroup$
    – littleO
    Dec 16 '18 at 20:56




















  • $begingroup$
    Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:04






  • 1




    $begingroup$
    How large is $A$? Is it sparse or does it have any other structure?
    $endgroup$
    – littleO
    Dec 16 '18 at 9:16










  • $begingroup$
    That is right, Ax is vector! I edited the question.
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 9:59










  • $begingroup$
    The only thing we know is that A is positive semi definite! Nothing more is available
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:00






  • 1




    $begingroup$
    @LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
    $endgroup$
    – littleO
    Dec 16 '18 at 20:56


















$begingroup$
Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
$endgroup$
– littleO
Dec 16 '18 at 9:04




$begingroup$
Since $Ax$ is a vector, what do you mean by $Ax in [a,b]$?
$endgroup$
– littleO
Dec 16 '18 at 9:04




1




1




$begingroup$
How large is $A$? Is it sparse or does it have any other structure?
$endgroup$
– littleO
Dec 16 '18 at 9:16




$begingroup$
How large is $A$? Is it sparse or does it have any other structure?
$endgroup$
– littleO
Dec 16 '18 at 9:16












$begingroup$
That is right, Ax is vector! I edited the question.
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 9:59




$begingroup$
That is right, Ax is vector! I edited the question.
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 9:59












$begingroup$
The only thing we know is that A is positive semi definite! Nothing more is available
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 10:00




$begingroup$
The only thing we know is that A is positive semi definite! Nothing more is available
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 10:00




1




1




$begingroup$
@LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
$endgroup$
– littleO
Dec 16 '18 at 20:56






$begingroup$
@LinAlg I don't have much experience with L-BFGS, is L-BFGS-B able to handle the hard constraint that $Ax in [a,b]^m$? I've read that it can handle box constraints like $x in [a,b]^n$, but $Ax in [a,b]^m$ is more difficult.
$endgroup$
– littleO
Dec 16 '18 at 20:56












1 Answer
1






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oldest

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2












$begingroup$

One option which I think is promising is to formulate your problem as minimizing $f(x) + g(x) + h(Ax)$, where $f(x) = x^T A x$ and $g(x) = 0$ and $h(y) = I_{[a,b]}(y)$. (That is, $h$ is the convex indicator function of the set denoted here as $[a,b]$.) The function $f$ is differentiable and $g$ and $h$ have easy proximal operators, so you can solve this optimization problem using the PD3O method (which is a recent three-operator splitting method).



In this approach, we never have to solve a linear system involving $A$, which is nice because $A$ is large.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:01











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1 Answer
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1 Answer
1






active

oldest

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active

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active

oldest

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2












$begingroup$

One option which I think is promising is to formulate your problem as minimizing $f(x) + g(x) + h(Ax)$, where $f(x) = x^T A x$ and $g(x) = 0$ and $h(y) = I_{[a,b]}(y)$. (That is, $h$ is the convex indicator function of the set denoted here as $[a,b]$.) The function $f$ is differentiable and $g$ and $h$ have easy proximal operators, so you can solve this optimization problem using the PD3O method (which is a recent three-operator splitting method).



In this approach, we never have to solve a linear system involving $A$, which is nice because $A$ is large.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:01
















2












$begingroup$

One option which I think is promising is to formulate your problem as minimizing $f(x) + g(x) + h(Ax)$, where $f(x) = x^T A x$ and $g(x) = 0$ and $h(y) = I_{[a,b]}(y)$. (That is, $h$ is the convex indicator function of the set denoted here as $[a,b]$.) The function $f$ is differentiable and $g$ and $h$ have easy proximal operators, so you can solve this optimization problem using the PD3O method (which is a recent three-operator splitting method).



In this approach, we never have to solve a linear system involving $A$, which is nice because $A$ is large.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:01














2












2








2





$begingroup$

One option which I think is promising is to formulate your problem as minimizing $f(x) + g(x) + h(Ax)$, where $f(x) = x^T A x$ and $g(x) = 0$ and $h(y) = I_{[a,b]}(y)$. (That is, $h$ is the convex indicator function of the set denoted here as $[a,b]$.) The function $f$ is differentiable and $g$ and $h$ have easy proximal operators, so you can solve this optimization problem using the PD3O method (which is a recent three-operator splitting method).



In this approach, we never have to solve a linear system involving $A$, which is nice because $A$ is large.






share|cite|improve this answer









$endgroup$



One option which I think is promising is to formulate your problem as minimizing $f(x) + g(x) + h(Ax)$, where $f(x) = x^T A x$ and $g(x) = 0$ and $h(y) = I_{[a,b]}(y)$. (That is, $h$ is the convex indicator function of the set denoted here as $[a,b]$.) The function $f$ is differentiable and $g$ and $h$ have easy proximal operators, so you can solve this optimization problem using the PD3O method (which is a recent three-operator splitting method).



In this approach, we never have to solve a linear system involving $A$, which is nice because $A$ is large.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 16 '18 at 9:12









littleOlittleO

29.8k646109




29.8k646109












  • $begingroup$
    Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:01


















  • $begingroup$
    Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
    $endgroup$
    – Majid Mohammadi
    Dec 16 '18 at 10:01
















$begingroup$
Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 10:01




$begingroup$
Thanks for response! I have seen some three operator method using different proximal gradient! Can you a bit explain the method?
$endgroup$
– Majid Mohammadi
Dec 16 '18 at 10:01


















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